reserve x,y,X for set,
  i,j,k,m,n for Nat,
  p for FinSequence of X,
  ii for Integer;

theorem Th1:
  for p being FinSequence,x being set holds not x in rng p & p is
  one-to-one iff p^<*x*> is one-to-one
proof
  let p be FinSequence,x be set;
A1: rng <*x*> = {x} by FINSEQ_1:38;
  rng p misses rng <*x*> & p is one-to-one & <*x*> is one-to-one iff p ^
  <*x*> is one-to-one by FINSEQ_3:91;
  hence thesis by A1,FINSEQ_3:93,ZFMISC_1:48,50;
end;
